| We investigate the differentiability of the Ferrand metric-density and find a sufficient condition that implies that the Ferrand metric-density is not differentiable and that the Ferrand Gaussian curvature is negative infinity at a point. We prove that a closed Euclidean ball, closed half-space, or the complement of an open Euclidean ball contained in a domain is convex with respect to the Ferrand metric. As a result, the arc-length parametrization of a Ferrand geodesic has Lipschitz continuous first derivative, and this result is sharp. We consider the Ferrand geometry of a convex domain. We show that Ferrand geodesics are unique, that Ferrand balls are strictly Euclidean convex, and that Ferrand geodesics can be prolonged to Ferrand geodesic rays. Finally, we study the notion of a rolling disk condition and provide a sufficient geometric condition which guarantees that the ratio of the hyperbolic and quasihyperbolic metric-densities, and also the ratio of the Ferrand and quasihyperbolic metric-densities, both approach one as an interior point approaches a boundary point. |
